Psoriasis is a chronic skin disease in which the process of hyper-proliferation (excessive division) of skin cells starts. Externally, psoriasis appears as red papules, on the surface of which there are scales of white–gray color. There is substantial evidence that T-helper cells take vital accountability for creating the hyper-proliferation of keratinocytes (skin cells), which causes itching of skin patches. In this paper, we propose a mathematical model describing the concentrations of T-helper and keratinocyte cell populations to predict cellular behaviors for psoriasis regulation under normal or anomalous immune circumstances. Local and global asymptotic stabilities of the model equilibria are investigated. Additionally, by introducing two scalar bounded controls into the model, the effect of combined immunotherapy using IL-10 and IL-22 inhibitors is analyzed. The optimal control problem of minimizing the cost of immune therapy and simultaneous optimizing the effect of this therapy on T-helper cells and keratinocytes proliferation is formulated and solved by applying the Pontryagin maximum principle. Within the restrictions of the proposed model, the obtained analytical and numerical outcomes suggest that the optimal strategy of injecting IL-10 and IL-22 inhibitors can be effective for psoriasis treatment.

中文翻译：

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引入 IL-10 和 IL-22 抑制剂的银屑病最佳免疫治疗模型

牛皮癣是一种慢性皮肤病，皮肤细胞开始过度增殖（过度分裂）。银屑病外观呈红色丘疹，表面有白灰色鳞片。有大量证据表明，T 辅助细胞对产生角质形成细胞（皮肤细胞）的过度增殖至关重要，这会导致皮肤斑块发痒。在本文中，我们提出了一个数学模型，描述了 T 辅助细胞和角质形成细胞群的浓度，以预测正常或异常免疫环境下银屑病调节的细胞行为。研究了模型平衡的局部和全局渐近稳定性。此外，通过在模型中引入两个标量有界控件，分析了使用 IL-10 和 IL-22 抑制剂联合免疫治疗的效果。应用Pontryagin最大值原理制定并解决了最小化免疫治疗成本并同时优化该疗法对T辅助细胞和角质形成细胞增殖的效果的最优控制问题。在所提出模型的限制范围内，获得的分析和数值结果表明，注射 IL-10 和 IL-22 抑制剂的最佳策略可有效治疗银屑病。